I agree with pretty much everything that has been said on fom recently in
defence of the reverse maths programme. But I'd like to throw into the
discussion of the significance of reverse mathematics a possible application
of reverse mathematics methodology to some issues concerning the role of
applied mathematics in science.
Many mathematical realists in the philosophy community justify their realism
about mathematical entities by citing the fact that mathematics is integral
and seemingly indispensable to the formulation and development of serious
theoretical science: e.g., quantum theory and relativity theory. This
argument was invented by W.V. Quine in the 1940s, and further amplified by
Hilary Putnam in the 1970s, and has come to be called The Quine-Putnam
Indispensability Argument (for mathematical realism).
For example, if you're a scientific realist and think that a
well-corroborated scientific theory like Quantum Electrodynamics or General
Relativity is *true* (or a very close approximation to the truth), then you
cannot seriously claim that the mathematics needed to formulate and develop
such theories is also *not true*. Such theories certainly include arithmetic
and much analysis, and there may be a certain amount of set theory required
(the set theory is a set theory with ur-elements, but it's still a set
theory nonetheless). For example, you cannot make sense of co-ordinate
charts, geodesics, field functions, tensor fields, gauge transformations,
fibre bundles, spacetime manifolds, etc., unless you believe that there are
such things as numbers, sets and functions.
In short, this kind of argument provides a genuine empirical justification
for the adoption of a realist view of mathematics, which Harvey Fiedman
described (on 13 Sept) thus:
=> There are a number of people who have a very strong Platonist or
realist
=> view, and strongly feel that they are uncovering an objective
reality
=> similar in spirit to material reality and physicists.
There is no claim in this Quinean argument that mathematical axioms are
(known to be true) a priori. The adoption of, say, certain existence axioms
of set theory is justified via a broad "holistic" argument concerning the
indispensability of such axioms to the development of serious scientific
theory, which itself is ultimately tested by experiment or observation. It
is quite conceivable, given Godel's results, that the addition of highly
abstract set existence axioms *could* make a difference to what a scientific
theory predicts about the physical world, and for these predictions be
empirically measureable.
So an important question to ask is to see exactly what mathematical axioms
are *needed* to prove important theorems required (already proved
informally) in theoretical physics. There are at least two examples that
have been recently discussed in the literature by the philosopher Geoffrey
Hellman:
[1] Gleason's Theorem: every state on a Hilbert space of dim > 2 is
regular.
[2] Hawking-Penrose Singularity Theorem(s): certain spacetime
structures (M, g, T) satisfying Einstein's field equations plus further
conditions must contain an initial singularity.
Hellman claims that there cannot be constructive proofs of these theorems.
There are many other mathematical results that could be investigated
similarly. E.g., Weyl's Theorem that the spectra of the position Q and
momentum P operators on Hilbert space, which satisfy the commutation
relation [Q, P] = ih, must be the whole real line.
So, an interesting question is: what is the reverse mathematics status of
Gleason's Theorem, Hawking-Penrose Theorems, etc.? What are the weakest set
existence axioms require to prove them: do we require non-constructivity,
impredicativity, etc.?
It would be striking, and of some philosophical significance (concerning the
justification for mathematics: our reasons for thinking that set theory is
*true*, and describes, if only partially, an abstract reality existing "out
there"), if it could be demonstrated that the mathematics needed for
standard accepted well-corroborated and tested theories from theoretical
physics was non-constructive or impredicative or what-have-you.
For a rather speculative example, the following sort of scenario is
conceivable. Suppose you're in a spaceship falling into a black hole and you
know from a theorem of Hawking that if you do X (say, navigate the booster
rockets in a certain way) then you will be saved, but the mathematics
*needed* to prove this theorem (if you do X then you'll be safe) is
demonstrably non-constructive or impredicative, then only a mathematical
realist could in good conscience act on this information. A constructivist
or predicativist would end up being sliced to shreds, as a result of his
disbelief in (the truth of) non-constructive or impredicative mathematics.
I don't know the answers to these questions, but it seems to me that reverse
mathematics is exactly the right methodology for studying what mathematics
really is indispensable to the formulation, and development, of some of our
standard scientific theories of the physical world. So this is another sense
in which the reverse mathematics programme contains a powerful set of ideas
for dealing with matters of what Friedman calls "general intellectual
interest", concerned with the vital philosophical question of whether we
should think of mathematics as aiming at undercovering the truth (realism),
or as mere manipulation of meaningless symbols (formalism) or something in
between (constructivism).
Jeff Ketland
j.j.ketland at lse.ac.uk
Department of Philosophy
London School of Economics
0171 955 6822